MUBO 11044 AUTHOR’S PROOF...PDF-OUTPUT AUTHOR’S PROOF Journal ID: 11044, Article ID: 9070, Date: 2007-06-28, Proof No: 1 UNCORRECTED PROOF « MUBO 11044 layout: Small Condensed

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Journal ID: 11044, Article ID: 9070, Date: 2007-06-28, Proof No: 1

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« MUBO 11044 layout: Small Condensed v.1.2 reference style: mathphys file: mubo9070.tex (Aiste) aid: 9070 doctopic: OriginalPaper class: spr-small-v1.1 v.2007/06/28 Prn:28/06/2007; 15:52 p. 1»

Multibody Syst DynDOI 10.1007/s11044-007-9070-6

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Effect of the centrifugal forces on the finite elementeigenvalue solution of a rotating blade: a comparativestudy

Luis G. Maqueda · Olivier A. Bauchau ·Ahmed A. Shabana

Received: 28 November 2006 / Accepted: 21 May 2007© Springer Science+Business Media B.V. 2007

Abstract In this study, the effect of the centrifugal forces on the eigenvalue solution ob-tained using two different nonlinear finite element formulations is examined. Both formula-tions can correctly describe arbitrary rigid body displacements and can be used in the largedeformation analysis. The first formulation is based on the geometrically exact beam the-ory, which assumes that the cross section does not deform in its own plane and remainsplane after deformation. The second formulation, the absolute nodal coordinate formula-tion (ANCF), relaxes this assumption and introduces modes that couple the deformation ofthe cross section and the axial and bending deformations. In the absolute nodal coordinateformulation, four different models are developed; a beam model based on a general contin-uum mechanics approach, a beam model based on an elastic line approach, a beam modelbased on an elastic line approach combined with the Hellinger–Reissner principle, and aplate model based on a general continuum mechanics approach. The use of the general con-tinuum mechanics approach leads to a model that includes the ANCF coupled deformationmodes. Because of these modes, the continuum mechanics model differs from the modelsbased on the elastic line approach. In both the geometrically exact beam and the absolutenodal coordinate formulations, the centrifugal forces are formulated in terms of the elementnodal coordinates. The effect of the centrifugal forces on the flap and lag modes of the rotat-ing beam is examined, and the results obtained using the two formulations are compared fordifferent values of the beam angular velocity. The numerical comparative study presentedin this investigation shows that when the effect of the ANCF coupled deformation modesis neglected, the eigenvalue solutions obtained using the geometrically exact beam and theabsolute nodal coordinate formulations are in a good agreement. The results also show thatas the effect of the centrifugal forces, which tend to increase the beam stiffness, increases;the effect of the ANCF coupled deformation modes on the computed eigenvalues becomes

L.G. Maqueda · A.A. Shabana (�)Department of Mechanical Engineering, University of Illinois at Chicago, 842 West Taylor Street,Chicago, IL 60607, USAe-mail: [email protected]

O.A. BauchauDaniel Guggenheim School of Aerospace Engineering, Georgia Institute of Technology,270 Ferst Street, Atlanta, GE 30332-0150, USA

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less significant. In the geometrically exact beam models, the two components of the normalstrains associated with the deformation of the cross section are assumed to be zero, and as aconsequence, the Poisson ratio effect is not considered. It is shown in this paper that whenthe effect of the Poisson ration is neglected, the eigenvalue solution obtained using the ab-solute nodal coordinate formulation and the general continuum mechanics approach is in agood agreement with the solution obtained using the geometrically exact beam model.

Keywords ???

1 Introduction

The dynamics of rotating beams has been the subject of a large number of investigations be-cause of its relevance to important engineering applications such as helicopters’ blades. Oneof the earliest studies is the work of Schilhans [14], who presented the partial differentialequation of the flexural vibration of a rotating beam in the steady state. The early investi-gations were concerned with one-dimensional beam in steady state motion where neitherthe Coriolis effect nor the coupling between extension and bending were considered. Thesignificant effect of the coupling between the beam extension and bending was recognized,and it was demonstrated that the neglect of the effect of the geometric centrifugal stiffeningthat results from this coupling leads to incorrect solutions. Johnson [10] documented theneed for considering the effect of the coupling between both flap and lag motions and theextensional motion of rotating blades. Wu and Haug [19] used substructuring techniques tomodel the coupling between the axial and flexural displacements, but no quantitative studywas given to determine the relation between the critical speed and the size of the substruc-ture at which instability may occur. The centrifugal geometric stiffening effect has beenalso accounted for by considering a prestressed reference configuration of the flexible body[12, 18]. Garcia-Vallejo et al. [6, 7] considered the geometric stiffening effect of a rotatingbeam when using the floating frame of reference formulation and the absolute nodal coor-dinate formulation (ANCF) [3]. They introduced a new correction to the floating frame ofreference formulation that leads to coupling between the bending and extension.

In most existing finite element beam models, the dimensions of the cross section areassumed to remain constant when the beam deforms. This, in fact, is the underlying as-sumption used in both Euler–Bernoulli and Timoshenko beam theories. In some of the finiteelement absolute nodal coordinate formulation beam models, this assumption is relaxedallowing the beam cross section dimensions to change. In the absolute nodal coordinate for-mulation, which allows for the use of general constitutive equations and strain-displacementrelationships, several methods can be used to formulate the elastic forces. One method thathas been used by several investigators is based on the general continuum mechanics ap-proach. When the general continuum mechanics approach is used, the resulting new beammodel leads to a geometric coupling between the deformation of the cross section and thebeam axial and bending deformations. This kinematic coupling can be important in the caseof very flexible structures and/or in some plasticity applications in order to realistically ac-count for the change in the cross section dimensions when the structures deform. However,in the case of very stiff and thin structures, the resulting ANCF coupled deformation modes[9, 15] can have high frequencies that do not compare well with the analytical solution thatis based on the assumption that the cross section dimensions do not change when the beamdeforms. Nonetheless, using the absolute nodal coordinate formulation, one can still obtainthe analytical eigenvalue solution by systematically eliminating the coupling between the

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cross section deformation and the beam axial and bending deformation [15]. In this case,another method for formulating the elastic forces must be used. Two of these methods areconsidered in this study, which is focused on examining the effect of the centrifugal forceson the eigenvalue solution of rotating beams. In the first of these two methods, the elasticline approach is used; while in the second method, the elastic line approach is used withthe Hellinger–Reissner principle. Multi-field principles are used often in the finite elementliterature to solve the locking problems.

It is the purpose of this study to examine the effect of the centrifugal forces on the finiteelement eigenvalue solution of rotating beams. The eigenvalue solution is obtained usingtwo different nonlinear finite element formulations. The first formulation employs a geo-metrically exact beam model which assumes that the cross section does not deform in itsown plane and remains plane after deformation, while the second formulation, the absolutenodal coordinate formulation, relaxes this assumption. Several ANCF beam models are usedin this study. The first is based on a general continuum mechanics approach, the second isbased on an elastic line approach, the third combines the elastic line approach with theHellinger–Reissner principle, and the fourth is a general continuum mechanics based platemodel for the beam. The centrifugal forces are formulated as function of the finite elementnodal coordinates and the beam angular velocity. The resulting frequencies of the flap andlag modes are determined, and the solutions obtained using the two different nonlinear finiteelement formulations are compared. It is shown that when the ANCF coupled deformationmodes are neglected [9, 15], the two solutions obtained using the two nonlinear formula-tions are in a very good agreement for different values of the beam angular velocity. It isimportant also to recognize that since the geometrically exact beam theory assumes thatthe beam cross section remains rigid, the two normal strain components associated with thecross section deformation are equal to zero; and as a consequence, Poisson ratio does notenter into the formulation of the elastic forces. It is shown in this investigation that when theeffect of the Poisson ratio is neglected, the eigenvalue solution obtained using the absolutenodal coordinate formulation and the general continuum mechanics approach is in a goodagreement with the solution obtained using the geometrically exact beam model. This paperis organized as follows. In Sect. 2, the geometrically exact beam formulation used in thisstudy is briefly discussed. In Sect. 3, the formulation of the centrifugal forces based on thegeometrically exact beam model is described. In Sect. 4, the four different finite element ab-solute nodal coordinate formulation beam models used in this study are presented. Section 5describes the formulation of the centrifugal forces based on the absolute nodal coordinateformulation. In this section, the eigenvalue equations expressed in terms of the absolutenodal coordinates are also presented. In Sect. 6, numerical results are presented in order toexamine the effect of the centrifugal forces on the eigenvalue solution of the rotating beam.The results obtained using the two nonlinear finite element formulations for the flap and lagmodes are compared for different beam angular velocities. Summary and conclusions drawnfrom this study are presented in Sect. 7.

2 Geometrically exact beam theory

In this section and the following section, the geometrically exact finite element beam theoryused in this investigation to formulate the centrifugal forces and study their effect on theeigenvalue solution is presented. More discussions on this formulation can be found in theliterature [1]. Application of this formulation to rotorcraft problems was demonstrated byBauchau et al. [2].

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2.1 Kinematics of the problem

Consider a beam of length L with a cross section A of arbitrary shape, as depicted in Fig. 1.The volume of the beam is generated by sliding the cross section along the reference line ofthe beam, which is an arbitrary space curve. For this particular application, the reference lineis selected to be coincident with the centerline of the beam. An inertial frame of referenceI = (i1, i2, i3) is used. Let X0(α1) be the position vector of a point on the centerline ofthe beam in the reference configuration; α1 is a curvilinear coordinate that measures lengthalong the beam centerline. The position vector of a material point on the beam can be writtenas

X(α1, α2, α3) = X0(α1) + �X(α1, α2, α3) (1)

where �X = α2E2(α1) + α3E3(α1). The vectors E2 and E3 define the plane of the crosssection of the beam, and α2 and α3 are material coordinates along those axes. The coordi-nates α1, α2 and α3 form a natural choice of coordinates (parameters) to represent the beam.Let the tangent to the centerline of the beam be E1 = ∂X0/∂α1. Since the cross section isperpendicular to the centerline at the reference configuration, and E2 and E3 are chosen tobe in the cross section plane and perpendicular to each other, the basis B0 = (E1,E2,E3) isorthonormal. This basis can be related to the inertial frame through a finite rotation tensorR0 such that Ei = R0ii . The derivative of this basis along the axis of the beam is

RT0 E′

i = Kii (2)

where ′ indicates a derivative with respect to α1, and K = RT0 R′

0 is the skew symmet-ric matrix associated with the curvature vector in the reference configuration. If K =[K1 K2 K3 ]T is the curvature vector, K1 is the twist, or pretwist, of the beam, andK2 and K3 its natural curvatures. The base vectors in the reference configuration areGi = ∂X/∂αi with i = 1,2,3, which can be written as

G1 = R0

[i1 + K(α2i2 + α3i3)

], G2 = R0i2, G3 = R0i3. (3)

Fig. 1 Geometrically exactbeam theory

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The metric tensor of the reference configuration can be readily computed as

G =[

(1 − α2K3 + α3K2)2 + (α3K1)

2 + (α2K1)2 −α3K1 α2K1

−α3K1 1 0α2K1 0 1

](4)

where each of the terms is calculated using Gij = GTi Gj with i, j = 1,2,3.

In the deformed configuration of the beam, the position vector of a material point iswritten as

r(α1, α2, α3) = r0(α1) + �r(α1, α2, α3) = X0(α1) + u(α1) + �r(α1, α2, α3) (5)

where r0 is the position of a material point on the centerline of the beam expressed as thesum of the position vector X0 of this point in the reference configuration and u, the center-line displacement vector. �r is now the position vector of a material point with respect tothe centerline in the deformed configuration. The base vectors in the deformed configurationbecome gi = ∂r/∂αi , and at the centerline ei = gi (α2 = α3 = 0), with i = 1,2,3. Two fun-damental assumptions are made concerning the deformation of the beam: the cross sectiondoes not deform in its own plane, and the cross section remains plane after deformation.Note that the plane of the cross section is not assumed to remain normal to the centerlineof the beam, allowing for transverse shearing deformations. These assumptions imply thateach cross section displaces and rotates like a rigid body. Consequently, vectors e2 and e3

remain mutually orthogonal, unit vectors. An orthonormal basis B = (j1, j2, j3) is defined asfollows: j2 = e2, j3 = e3, j1 = j2 × j3. Note that e1 is not a unit vector, nor is it orthogonal toe2 or e3, as axial and transverse shearing strains develop during deformation. Let R(α1) bethe finite rotation tensor that brings basis B0 to basis B , that is, j1(α1) = R(α1)E1 = RR0i1.Since the cross section remains rigid during the deformation process, the vector �r must beentirely contained in the plane of the cross section in the deformed configuration. Hence,

�r(α1, α2, α3) = α2e2 + α3e3 = α2j2(α1) + α3j3(α1) = R(α1)�X (6)

which implies that during deformation, the relative position vector of all material points ofa cross section undergo a rigid body rotation defined by the finite rotation tensor R. Theposition vector of a material point in the deformed configuration can be written as

r(α1, α2, α3) = X0(α1) + u(α1) + R(α1)�X(α1, α2, α3). (7)

The base vectors in the deformed configuration can be readily obtained as g1 = E1 + u′ +α2e′

2 + α3e′3, g2 = e2 and g3 = e3; the corresponding base vectors at the centerline are then

e1 = E1 + u′, e2 = j2 and e3 = j3, respectively. The components of vectors e1, e2, and e3 canbe expressed in the material system B as e∗

i = (RR0)T ei , where ∗ means that the vector is

defined in the material coordinate system. The vector components e∗1, e∗

2, and e∗3 are denoted

e∗1 = [1 + e11 2e12 2e13 ]T , e∗

2 = [0 1 0 ]T , e∗3 = [0 0 1 ]T (8)

where e11, e12, and e13 are strain parameters. Vectors e∗2 and e∗

3 are orthonormal due to theassumption that the cross section displaces and rotates like a rigid body. The components ofvectors e′

2 and e′3 can also be expressed in the material system, B , as (e′

2)∗ = (RR0)

T e′2 =

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(RR0)T j′2 = ki2, (e′

3)∗ = (RR0)

T e′3 = (RR0)

T j′3 = ki3, where the components of the curva-ture are defined as the elements of the following matrix:

k = (RR0)T (RR0)

′ =[ 0 −k3 k2

k3 0 −k1

−k2 k1 0

]. (9)

Next, the components of the base vectors in the deformed configuration are expressed in thematerial system B as g∗

i = (RR0)T gi , and their components are found as

g∗1 = [1 + e11 − α2k3 + α3k2 2e12 − α3k1 2e13 − α2k1 ]T

g∗2 = [0 1 0 ]T , g∗

3 = [0 0 1 ]T}

. (10)

2.2 Strain analysis

The Green–Lagrange strain components εij are defined as

εij = 1

2(gij − Gij ) (11)

where gij = gTi gj is the metric tensor in the deformed configuration. The strain compo-

nents are defined in the curvilinear coordinate system defined by the coordinates α1, α2, α3.However, it is more convenient to work in a locally rectangular coordinate system definedby the basis B0. The strain components expressed in these two systems are related byεij = εpq

∂αp

∂αi

∂αq

∂αj, where the rectangular coordinates along E1, E2, and E3 are denoted α1,

α2 and α3, respectively. It follows that

∂αi

∂αj

=[ √

G 0 0−α3K1 1 0α2K1 0 1

](12)

where√

G = (1 − α2K3 + α3K2) is the square root of the determinant of the metric tensorin the reference configuration. The strain components in the rectangular system become

Gε11 = ε11 + 2α3K1ε12 − 2α2K1ε13√

Gε12 = ε12,√

Gε13 = ε13

ε22 = ε33 = ε23 = 0

⎫⎪⎪⎬⎪⎪⎭ . (13)

The fact that the strains in the plane of the cross section vanish is a direct implication ofassuming undeformable cross section.

The formulation presented in this section focuses on beams with a shallow curvature,that is, α2K3 � 1 and α3K2 � 1, which implies

√G ≈ 1. It is also assumed that the beam

undergoes small deformations, that is, all the strain and curvature components are assumedto remain much smaller than unity: e11, 2e12, 2e13, α2κ1, α3κ1, α3κ2 and α2κ3 � 1, whereκi = ki −Ki are the elastic curvatures. Using these assumptions together with (11), the straincomponents of (13) then become

ε11 = e11 − α2κ3 + α3κ2

2ε12 = 2e12 − α3κ1

2ε13 = 2e13 + α2κ1

⎫⎪⎪⎬⎪⎪⎭ . (14)

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These equations can be written in a matrix form as follows

ε = e + �XT κ (15)

where ε = [ε11 2ε12 2ε13]T and e = [e11 2e12 2e13]T .The reference curve strains are

[1 + e11 2e12 2e13 ]T = (RR0)T e1, κ = k − K = RT

0

(RT R′)R0 − RT

0 R0. (16)

The base vectors are e1 = E1 + u′, e2 = RR0i2 and e3 = RR0i3. Equation (14) definesthe strain-displacement relationships. The reference axis strains and curvatures are definedby (16). The strains are expressed in terms of six displacement components, three trans-lational components, u, and the three rotational components that define the finite rotationtensor R.

3 Eigenvalue solution using the geometrically exact formulation

In this section, the governing equations of the rotating beam using the geometrically exactbeam formulation and the constitutive laws are presented.

3.1 Governing equations

First, the governing equations for the static problem are presented. The principle of virtualwork states ∫ L

0

∫A

δεT τ ∗ dAdα1 = δWext (17)

where τ ∗ = [τ ∗11 τ ∗

12 τ ∗13]T is the stress vector and δε = δe + �XT δκ . After integration over

the cross section of the beam, (17) becomes

∫ L

0

(δeT N∗ + δκT M∗)dα1 = δWext (18)

where N∗ = [N∗1 N∗

2 N∗3 ]T = ∫

Aτ ∗ dA and M∗ = [M∗

1 M∗2 M∗

3 ]T = ∫A

�Xτ ∗ dA are theaxial and transverse forces, and the twisting and bending moments, respectively, measuredin the material frame. Using (16), the variations in strain components can be expressed as

δe = (RR0)T(δu′ + eT

1 δψ), δκ = (RR0)

T δψ ′ (19)

where δψ = RT δR is the virtual rotation vector. The principle of virtual work becomes

∫ L

0

[(δu′T + δψT e1

)RR0N∗ + δψ ′T RR0M∗]dα1 = δWext. (20)

The beam internal forces and moments in the inertial system, N = RR0N∗ and M = RR0M∗,respectively, are defined. The virtual work of the external forces can be written as

δWext =∫ L

0

(δuT Fe + δψT Me

)dα1 (21)

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where Fe and Me are the external loads and moments on the beam per unit of length of thereference configuration. Integrating by parts and invoking the arbitrary nature of displace-ment variations yields the governing equations of the problem as

N′ = −Fe, M′ + e1N = −Me. (22)

In the case of the dynamic problem, the inertia forces need to be introduced in the governingequations. In order to calculate the inertia forces, one needs to define the variation of kineticenergy. The inertial velocity v of a material point can be found by taking a time derivativeof the inertial position vector defined in (7), to find v = u + R�X. The components of theinertial velocity vector expressed in the material frame then become

v∗ = (RR0)T v = (RR0)

T u + (RR0)T �XT ω (23)

where ω is the angular velocity. The two terms in (23) are associated with translation androtation of the cross section respectively. Variations of the velocity components are then

δ[(RR0)

T u] = (RR0)

T(δu + ˜uT

δψ), δ

[(RR0)

T ω] = (RR0)

T δψ . (24)

The sectional velocities in the material system are

υ∗ = [(RR0)

T u (RR0)T ω

]T. (25)

The governing equations of motion of the problem are found with the help of Hamilton’sprinciple as

H − N′ = Fe, L + ˜uH − M′ − e1N = Me (26)

where H and L are the sectional linear momentum and angular momentum, respectively.

3.2 Constitutive equations

To complete the formulation, the constitutive laws of the beam must be specified. First, thestiffness characteristics of the section are defined by the following relationship written in thematerial frame:

[ e κ ]T = C∗[N∗ M∗ ]T(27)

where C∗ is the fully populated, 6 × 6 stiffness matrix of the section. This matrix can be ob-tained from the variational asymptotic procedure described by Hodges [8] and Cesnik et al.[4]. This approach provides a rigorous proof of the fact that the original three-dimensionalelasticity problem splits into two independent problems: a linear, two-dimensional analy-sis of the cross section of the beam, and a nonlinear, one-dimensional problem along theaxis of the beam. The first problem takes into account the three-dimensional nature of thebeam deformation and includes a full finite element discretization of the section, which canhandle beam of arbitrary cross section made of anisotropic materials. The material framedefined in the preceding section is, in fact, a floating frame of reference for which the av-erage warping vanishes. This removes the kinematic assumption stated above. The second,one-dimensional problem along the axis of the beam is identical to the geometrically exactformulation described above. The variational asymptotic approach to the problem guaran-tees the convergence of the results to those of three-dimensional elasticity.

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The inertial constitutive laws relate the sectional linear momentum and angular momen-tum to the sectional velocities,

[H∗ L∗ ]T = μ∗υ∗. (28)

The sectional mass matrix in the material system is

μ∗ =[

m mη∗T

mη∗ I∗0

](29)

where m is the mass of the beam per unit span, η∗ the components of position vector ofthe center of mass of the section with respect to the reference line of the beam, and I∗

0 thecomponents of the sectional mass moment of inertial tensor. Both η∗ and I∗

0 are measured inthe material frame.

3.3 Determination of natural frequencies

To determine the natural frequencies of a rotating structure, a two-step procedure is fol-lowed. First, the static equilibrium configuration of the beam under the steady centrifugalforces associated with the rotation of the beam is computed. Next, the equations of motionof the system are linearized about this equilibrium configuration to obtain the linearizedmass and stiffness matrices, which, in turns, form a generalized eigenvalue problem for theevaluation of the natural frequencies and associated mode shapes.

4 Absolute nodal coordinate formulation models

As previously mentioned, the goal of this study is to compare between the eigenvalue resultsobtained using the geometrically exact beam theory and the absolute nodal coordinate for-mulation when the effect of the centrifugal forces is considered. In this section, the differentfinite element absolute nodal coordinate formulation models used in this study are reviewed.Beam and plate finite element models are considered. While for the plate element model,only the general continuum mechanics approach is used to formulate the elastic forces, sev-eral models are used with the beam element. Specifically, three different beam models areconsidered; the first is based on the general continuum mechanics approach [16, 17, 20], thesecond is based on the elastic line approach [15, 17], while the third combines the elasticline approach and the Hellinger–Reissner principle [15]. In this section, the beam elementwith the three elastic force models is first presented, followed by the plate element modelthat employs the continuum mechanics to formulate the elastic forces [11].

In the absolute nodal coordinate formulation, the global position vector r of an arbitrarypoint on the finite element (beam or plate) can be defined using the element shape functionsand the nodal coordinate vector as follows:

r = S(x, y, z)e (30)

where S is the element shape function matrix expressed in terms of the element spatial co-ordinates x, y, and z; and e is the vector of nodal coordinates that consist of absolute nodalposition coordinates and position coordinate gradients. The element shape function used in(30) for the beam and plate elements used in this study are presented in Appendix 1. The use

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Fig. 2 Absolute nodalcoordinate formulation beammodel

of the representation in (30) leads to a constant mass matrix and zero Coriolis and centrifu-gal forces associated with the absolute nodal coordinates. In order to compare the ANCFeigenvalue solution with the results obtained using the geometrically exact beam model, theequations of motion are defined in a rotating coordinate system, leading to nonzero centrifu-gal forces.

4.1 Beam element formulation

Figure 2 shows the global and local coordinate systems used to define the absolute posi-tion and gradient coordinates for the three-dimensional beam element. The element nodalcoordinate vector at node n is defined as follows:

en =[

rnT

(∂rn

∂x

)T (∂rn

∂y

)T (∂rn

∂z

)T]T

. (31)

The vector of the element nodal coordinates can be written as follows:

e = [eAT

eBT ]T. (32)

This element has 24 degrees of freedom; 12 degrees of freedom per node.

Continuum mechanics approach In the general continuum mechanics approach [20], theelastic forces are formulated using the Green–Lagrange strain tensor defined as

εm = 1

2

(JT J − I

)(33)

where J is the matrix of the position vector gradients, and I is the identity matrix. Using theprinciple of virtual work, the vector of generalized elastic forces Qe is defined as

Qe = −∫

V

(∂ε

∂e

)T

Eε dV (34)

where ε is a vector that consist of the six independent Green–Lagrange strain components,and E is the matrix of elastic coefficients [5]. The use of the continuum mechanics approachto formulate the elastic forces leads to a model that includes the ANCF coupled deforma-tion modes [9, 15]. While these modes, as previously mentioned, can be important in veryflexible structures and plasticity applications, for stiff and thin beams, the ANCF coupleddeformation modes can have very high frequencies that do not compare well with the an-alytical solutions that are obtained based on the small deformation assumptions and theassumption of the rigidity of the cross section.

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Elastic line approach In the elastic line approach [15], all the deformation modes aredefined along the beam centerline. The slopes on the centerline are defined as

rx = r,x(x,0,0), ry = r,y(x,0,0), rz = r,z(x,0,0) (35)

where r,α = ∂r/∂α with α = x, y, z. The strains and curvatures used in this model to for-mulate the elastic forces are defined as

εx = 1

2

(rTx rx − 1

), εy = 1

2

(rTy ry − 1

), εz = 1

2

(rTz rz − 1

),

γyz = rTy rz, γxy = rT

x ry, γxz = rTx rz,

κx = 1

2

(rTz rxy − rT

y rxz

), κy = rT

z rxx, κz = rTy rxx

⎫⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎭

(36)

where rxx = r,xx(x,0,0).In the elastic line approach, the total strain energy of the beam element We can be written

as the sum of four different strain energy terms as follows:

We = Wc + Ws + Wb + Wt (37)

where subscripts c, s, b and t refer, respectively, to extension and shear strain based on thedefinition of the vectors given in (35), shear strain, bending strain and torsion strain. Thesestrain energy terms are calculated as follows [15]:

Wc = 1

2

∫ l

0AεT Eε dx, Ws = 1

2

∫ l

0A(Gkyγxy + Gkzγxz) dx

Wb = 1

2

∫ l

0E

(Iyκ

2y + Izκ

2z

)dx, Wt = 1

2

∫ l

0kxGIpκx dx

⎫⎪⎪⎪⎬⎪⎪⎪⎭

(38)

where the integration is carried over the original length since small deformation assumptionsare used in this paper, and

ε = (εx, εy, εz, γyz)T , E = 2G

(1 − 2ν)

⎡⎢⎣

1 − ν ν ν 0ν 1 − ν ν 0ν ν 1 − ν 00 0 0 (1−2ν)

2

⎤⎥⎦ . (39)

In these expressions, l is the beam element length in the reference configuration, E is themodulus of elasticity, G is the modulus of rigidity, ν is the Poisson ratio, A is the crosssection area, Iy and Iz are the second moments of area, Ip is the polar moment of area,and kx , ky , and kz are shear factors. In the case of the elastic line approach, the vector ofgeneralized elastic forces Qe is defined as:

Qe = −∂We

∂e. (40)

In the elastic line approach, the geometric coupling between the cross section and bendingdeformations is not considered in the formulation of the elastic forces. Consequently, a beammodel based on this approach does not include the ANCF coupled deformation modes.

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Elastic line approach/Hellinger–Reissner principle Multi-field variational principles arefrequently used in the finite element literature to solve different locking problems, includingvolumetric, shear and membrane locking. In this section, as an example, the Hellinger–Reissner principle is used. In this principle, independent interpolation is made for the trans-verse shear stresses [13]. One can assume that the shear stresses will vary linearly over theelastic line of the element [15]. Both shear stresses τxy and τxz can be interpolated indepen-dent of the displacement field interpolation. For example, consider the τxy component. Thisshear strain component can be assumed in the following form:

τxy = Nτ ∗xy (41)

where N is an assumed shape function matrix, and τ ∗xy is a vector of shear stresses that can

be determined using the Hellinger–Reissner principle. Schwab and Meijaard [15] assumeda linear interpolation in (41). In this case, the shape function matrix is given by

N = [1 − ξ ξ ] (42)

where ξ = x/l. Using this linear interpolation, the vector of shear stresses τ ∗xy has two

elements and can be defined as

τ ∗xy = [

τAxy τB

xy

]T(43)

where τAxy = τxy (ξ = 0) and τB

xy = τxy (ξ = 1). Using the Hellinger–Reissner principle, thestrain energy associated with the shear strain in the xy-plane can be written as

W ∗xy =

∫V

(τxyγxy − Wc

xy(τxy))dV . (44)

In this equation, γxy is as defined in terms of the nodal coordinates using (36), and Wcxy is

the complementary shear stress energy which is defined by the equation

Wcxy(τxy) = 1

2Gky

(τxy)2. (45)

Substituting (41) and (45) into (44), we obtain

W ∗xy =

∫V

(τ ∗T

xy NT γxy − 1

2Gky

τ ∗Txy NT Nτ ∗

xy

)dV . (46)

By minimizing this strain energy with respect to τ ∗xy , the stress vector τ ∗

xy can be determinedin terms of γxy . In a similar manner τ ∗

xz can be determined. Using these vectors, the shearstresses τxy and τxz can be interpolated and used in the definition of the total strain energyfunction which can be used to define the elastic forces as previously described in this section.It is important to point out that the Hellinger–Reissner principle is a stress based variationalprinciple. Multi-field (mixed) strain based variational principles are also frequently used inthe finite element literature to deal with the locking problems.

4.2 Plate element model

Figure 3 shows the global and local coordinates used to define the absolute position andgradient coordinates for this element. The element nodal coordinate vector at node n is

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Fig. 3 Absolute nodalcoordinate formulation platemodel

defined as follows [11]:

en =[

rnT

(∂rn

∂x

)T (∂rn

∂y

)T (∂rn

∂z

)T]T

. (47)

The vector of the element nodal coordinates can be written as follows:

e = [eAT

eBTeCT

eDT ]T. (48)

This element has 48 degrees of freedom; 12 degrees of freedom per node. Using the generalcontinuum mechanics approach, the elastic forces can be formulated using (33) and (34);by following a procedure similar to the one used for the beam element. Therefore, thisplate element model also includes the ANCF coupled deformation modes. The plate elementshape function matrix used in this investigation is presented in Appendix 1 of this paper.

5 Formulation of the ANCF centrifugal forces

In this section, the method used in this investigation to study the effect of the centrifugalforces on the eigenvalue solution when using the absolute nodal coordinate formulation(ANCF) is presented. Recall that the mass matrix obtained using the absolute nodal coordi-nate formulation is constant, and as a result, the centrifugal and Coriolis forces are equal tozero. If a frame that rotates with the beam is used to define the coordinates instead of theinertial frame, the resulting equations will include centrifugal forces as will be demonstratedin this section. These forces can then be expressed in terms of the beam specified constantangular velocity.

The equation of motion of the rotating beam formulated using the absolute nodal coordi-nate formulation can be written as

Me + Qe = Qr (49)

where M is the constant symmetric mass matrix, e is the vector of nodal accelerations, Qe

is the vector of generalized elastic forces, and Qr is the vector of the constraint forces.In order to write the equations of motion in terms of coordinates defined with respect tothe rotating frame, the absolute coordinates need to be expressed in terms of the new setof coordinates. Assuming that the first node of the beam does not translate, the followingcoordinate transformation can be developed:

e = Aq (50)

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where q is the vector of absolute coordinates defined in the rotating frame, and the matrixA is the coordinate transformation matrix. In this study, a simple rotation of the beam abouta fixed axis is considered. In this case, without any loss of generality, the matrix A can bewritten in the case of the absolute nodal coordinate formulation as

A =

⎡⎢⎢⎣

A0 0 . . . 00 A0 . . . 0...

.... . .

...

0 0 . . . A0

⎤⎥⎥⎦ (51)

where A has dimension equal to the total number of nodal coordinates, and A0 is a 3 × 3orthogonal matrix defined as

A0 =[ cos θ − sin θ 0

sin θ cos θ 00 0 1

](52)

where θ is the angle of rotation about the fixed axis of rotation. Note that both matrices Aand A0 are orthogonal matrices.

Differentiating (50) twice with respect to time, substituting the result into (49), and pre-multiplying by AT , the equations of motion in the rotating frame can be written as

AT MAq + 2AT MAq + AT MAq + AT Qe = AT Qr (53)

where the matrices A and A are presented in Appendix 2. In the rotating beam model usedin the comparative numerical study presented in this investigation, simple linear constraintsare imposed on the absolute coordinates q, as will be discussed in the following section.Using these simple constraint equations, the vector of coordinates q can be written in termsof a selected set of independents coordinates qi as follows:

q = Bqi + γ (54)

where B and γ are constant in the application considered in this paper. Substituting thisequation into (53), one obtains the equations of motion of the system expressed in terms ofindependent coordinates defined in the rotating frame. These equations can be written as:

Mqi + Cqi + Kqi + Qγ + Qe = 0 (55)

where

M = BT AT MAB, C = 2BT AT MAB,

K = BT AT MAB, Qγ = BT AT MABγ

Qe = BT AT Qe

⎫⎪⎪⎬⎪⎪⎭ . (56)

Since the equations of motion are expressed in terms of the independents coordinates, theconstraint forces are automatically eliminated.

As pointed out by Garcia-Vallejo et al. [7], the forms of the mass matrix and the elasticforces in the absolute nodal coordinate formulation do not change under an orthogonal co-ordinate transformation. That is, AT MA = M for any orthogonal matrix A. It follows thatthe first matrix in (56) reduces to M = BT MB. While the transformation matrix A and itsderivatives appear in other components of (56), one can also show that the matrices and

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vectors that appear in this equation do not depend on the angle of rotation θ . To this end, thefollowing identities can be used:

AT0 A0 = I, AT

0 A0θ = I, A0θθ = −A0r (57)

where A0 is defined in (52), and

A0θ = ∂A0

∂θ, I =

[0 −1 01 0 00 0 0

], A0r =

[ cos θ − sin θ 0sin θ cos θ 0

0 0 0

]. (58)

Utilizing these identities and using the fact that the element shape function in the absolutenodal coordinate formulation can always be written in the following form:

S = [S1I S2I . . . SmI ], (59)

one can show, as demonstrated in Appendix 2, that the matrices and vectors that appearin (56) do not depend on the angle θ that defines the orientation of the rotating frame withrespect to the inertial frame. This important fact will be utilized in the development presentedin the remainder of this study.

The goal in this study, as previously mentioned, is to examine the effect of the centrifugalforces on the eigenvalue solution of the rotating beam about an equilibrium position. To thisend, the static equilibrium position coordinates qis are first determined for a given specifiedconstant value of the angular velocity. Other velocity and acceleration coordinates are as-sumed to be zero. Using qis to define the position coordinates in the equations of motion,one can formulate an eigenvalue problem, which can be solved for the flap and lag modesof the rotating beam.

5.1 Static equilibrium

In the static equilibrium analysis presented in this section, the effect of the gravity forcesis neglected, and the static configuration due to the effect of the centrifugal forces is deter-mined. As previously discussed in this section, the matrices that appear in the equations ofmotion of the rotating beam obtained using the absolute nodal coordinate formulation do notdepend on the angle that defines the orientation of the rotating frame with respect to the in-ertial frame. Consequently, the static equilibrium configuration qis of the model used in thisinvestigation does not depend on the orientation angle θ . In the case of the static analysis,qis = qis = 0, which upon the use of (55) becomes

Kqis + Qγ + Qe(qis ) = 0. (60)

This is a nonlinear system of equations that can be solved numerically using a Newton–Raphson algorithm to determine qis . In the preceding equation, the stiffness matrix K andthe centrifugal force vector Qγ depend on the prescribed angular velocity, which is assumedto be constant.

5.2 Eigenvalue analysis

In order to obtain the solution of the eigenvalue problem of the rotating beam including theeffect of the centrifugal forces, the equations of motion at a particular static configuration qis

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and a prescribed angular velocity ω0 = θ are linearized. The linearized equations of motionused in this investigation to perform the eigenvalue analysis are

Mqi + Cqi + K∗qi = 0 (61)

where

K∗ = K + BT AT ∂Qe

∂qi

(qis ). (62)

Note that in (61), there is a damping term. The effect of this damping term on the computedeigenvalues was found to be significant. It is also important to mention before concludingthis section that the procedure used for formulating the eigenvalue problem based on theabsolute nodal coordinate formulation is consistent with the procedure used in the geomet-rically exact beam theory previously discussed in this investigation.

6 Numerical results

In this section, the results obtained for the eigenvalue analysis of the rotating cantileverbeam using different nonlinear finite element absolute nodal coordinate formulations arepresented and compared with the results obtained using the geometrically exact beam model.The dimensions and material properties of the beam are presented in Table 1. The resultsare obtained in this study assuming that the beam is rotating with an angular velocity of0.1ω0,0.2ω0,0.3ω0, . . . ,ω0, where the value of ω0 is presented in Table 1. The main goalof this comparison is to show that when the effect of the ANCF coupled deformation modesis neglected, the eigenvalue solutions obtained using the geometrically exact beam theoryand the absolute nodal coordinate formulation are in good agreement when the effect of thecentrifugal forces of the rotating beam is taken into consideration in the linearized equations.

In the case of the beam model based on the ANCF, the degrees of freedom that areconstrained at the first node are:

r, ry |x, ry |z, rz|x (63)

where x|α indicates the α-th component of vector x, with α = x, y, z. Note that the resultingsimple constraints used in this model can be written in the form of (54). In the case of theplate model based on the ANCF, the same constraints as in (63) are used for the two nodesfixed at the base. The total number of elements used for the beam and plate models is ten.Therefore, the total number of degrees of freedom of the system is 126 and 252 for thebeam and plate models, respectively. In the case of the geometrically exact formulation, the

Table 1 Properties of therotating beam Parameter Symbol Value Units

Density ρ 2699.23 (kg/m3)

Young modulus E 727777 × 105 (N/m2)

Poisson ratio ν 0.3 –

Beam length L 8.178698 (m)

Beam width w 0.33528 (m)

Beam height h 0.033528 (m)

Angular velocity ω0 27.02 (rad/s)

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Table 2 Eigenfrequencies for the first flap mode at different angular velocities

Angular Geometri- ANCF_beam ANCF_beam ANCF_beam ANCF_plate

Veloc. cally exact (continuum (elastic line (elastic line + (continuum

beam mechanic approach) Hellinger– mechanic

model approach) Reissner) approach)

approach

0.0 ∗ ω0 2.64 3.07 2.64 2.64 2.83

0.1 ∗ ω0 3.95 4.25 3.95 4.05 4.09

0.2 ∗ ω0 6.37 6.59 6.38 6.44 6.49

0.3 ∗ ω0 9.00 9.19 9.02 9.06 9.12

0.4 ∗ ω0 11.67 11.86 11.71 11.73 11.81

0.5 ∗ ω0 14.35 14.55 14.41 14.43 14.51

0.6 ∗ ω0 17.04 17.26 17.12 17.16 17.22

0.7 ∗ ω0 19.74 19.98 19.85 19.88 19.94

0.8 ∗ ω0 22.44 22.70 22.57 22.61 22.67

0.9 ∗ ω0 25.13 25.42 25.30 25.34 25.40

1.0 ∗ ω0 27.83 28.15 28.03 28.05 28.13

Table 3 Eigenfrequencies for the second flap mode at different angular velocities




mode approach) Reissner approach)

approach)

0.0 ∗ ω0 16.55 19.46 16.76 16.61 17.97

0.1 ∗ ω0 17.92 20.65 18.13 18.30 19.25

0.2 ∗ ω0 21.52 23.86 21.71 21.86 22.66

0.3 ∗ ω0 26.42 28.40 26.62 26.73 27.40

0.4 ∗ ω0 32.01 33.73 32.22 32.33 32.89

0.5 ∗ ω0 37.96 39.51 38.21 38.30 38.77

0.6 ∗ ω0 44.12 45.56 44.41 44.55 44.89

0.7 ∗ ω0 50.40 51.78 50.74 50.86 51.16

0.8 ∗ ω0 56.76 58.11 57.16 57.33 57.51

0.9 ∗ ω0 63.17 65.51 63.64 63.80 63.94

1.0 ∗ ω0 69.62 70.98 70.17 70.22 70.41

model has 72 degrees of freedom. It was observed that if the number of degrees of freedomof the absolute nodal coordinate formulation model is reduced to half, the difference in theresults obtained is less than 4%. It is important, however, to point out that unlike other largedeformation finite element formulations, the absolute nodal coordinate formulation allowsfor using a relatively small number of elements in the case of very flexible structures.

Tables 2 and 3 show, respectively, the results corresponding to the first and second flapmodes for different values of the angular velocity. Table 4 shows the results for the first

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Table 4 Eigenfrequencies for the first lag mode at different angular velocities




model approach) Reissner approach)

approach)

0.0 ∗ ω0 26.34 30.54 25.34 25.22 26.72

0.1 ∗ ω0 26.41 30.69 25.38 26.29 26.77

0.2 ∗ ω0 26.49 30.76 25.45 26.37 26.90

0.3 ∗ ω0 26.61 30.87 25.57 26.50 27.12

0.4 ∗ ω0 26.79 31.03 25.74 26.68 27.43

0.5 ∗ ω0 27.02 31.23 25.95 26.91 27.81

0.6 ∗ ω0 27.29 31.48 26.20 27.20 28.27

0.7 ∗ ω0 27.60 31.76 26.48 27.52 28.79

0.8 ∗ ω0 27.94 32.07 26.80 27.88 29.38

0.9 ∗ ω0 28.32 32.42 27.14 28.27 30.02

1.0 ∗ ω0 28.72 32.80 27.52 28.66 30.70

lag mode. In the case of the first flap mode, the increase in the eigenfrequency due to thecentrifugal effect is about 1000% when the angular velocity is equal to ω0. For the secondflap mode, this increase is about 300%. On the other hand, in the case of the first lag mode,the increase is only about 8%. These significantly different effects of the centrifugal forceson the flap and lag modes are mainly attributed to the direction of the centrifugal forces.

The use of a general continuum mechanics approach to formulate the elastic forces leadsto a model that includes the ANCF coupled deformation modes, and consequently, leads tohigher values of the eigenfrequencies as explained by Schwab and Meijaard [15]. However,when the effect of the centrifugal forces becomes more significant, the effect of the coupleddeformation modes becomes less significant. The centrifugal forces tend to also increase thebeam stiffness.

The use of an elastic line approach, in which the effect of the ANCF coupled deforma-tion modes is eliminated, shows very good agreement with the solution obtained using thegeometrically exact beam model. The very small differences in the results can be attributedto differences in the implementation, numerical procedures and convergence criteria. Thenumerical results obtained in this study also show a very good agreement between the re-sults obtained using the two different nonlinear finite element formulations for higher orderflap and lag modes. Recall that these two large deformation finite element formulations areconceptually different and lead to different beam and plate models, as previously discussedin this study.

As previously mentioned in this paper, the geometrically exact beam theory assumes thatthe beam cross section remains rigid, and as a consequence, the two normal strain compo-nents associated with the cross section deformation are equal to zero. Therefore, Poissonratio does not enter into the formulation of the elastic forces when the geometrically exactbeam theory is used. Table 5 shows the solution obtained using the absolute nodal coordinateformulation and the general continuum mechanics approach when the effect of the Poissonratio is neglected and the angular velocity is equal to zero. It is clear from the results pre-sented in this table that the eigenvalue solution obtained using the absolute nodal coordinate

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901

902

903

904

905

906

907

908

909

910

911

912

913

914

915

916

917

918

919

920

921

922

923

924

925

926

927

928

929

930

931

932

933

934

935

936

937

938

939

940

941

942

943

944

945

946

947

948

949

950

Table 5 Effect of Poisson ratio

Geometrically exact ANCF_beam ANCF_beam

beam model (continuum mechanic (continuum mechanic

approach) approach)

ν = 0.3 ν = 0.0

1st flap 2.64 3.07 2.76

2nd flap 16.55 19.46 16.91

1st lag 26.34 30.54 26.45

formulation and the general continuum mechanics approach is in a good agreement with thesolution obtained using the geometrically exact beam model.

7 Summary and conclusions

In this study, the effect of the centrifugal forces on the eigenvalue solution obtained us-ing two different nonlinear finite element formulations was examined. Both formulationscan describe arbitrary rigid body motion and can be used for large deformation problems.The first formulation was based on a geometrically exact beam theory, which assumes thatthe cross section does not deform in its own plane and remains plane after deformation.The second formulation was based on the absolute nodal coordinate formulation (ANCF),which relaxes the assumption of the rigidity of the cross section. The results obtained in thisstudy showed that when the ANCF coupled deformation modes are eliminated, one obtainsa very good agreement between the solutions of the geometrically exact beam model andthe absolute nodal coordinate formulation models when the effect of the centrifugal forcesis considered. On the other hand, the use of a continuum mechanics approach to formulatethe elastic forces leads to kinematic coupling between the deformation of the cross sectionand the bending deformation. While this coupling can be significant in the case of veryflexible structures and/or plasticity problems, the coupled deformation modes tend to sig-nificantly increase the beam bending stiffness. The centrifugal forces also produce similareffect. As a consequence, as the effect of the centrifugal forces increases, the effect of theANCF coupled deformation modes decreases and becomes less significant, as demonstratedby the results presented in this paper.

Acknowledgements This work was supported by the US Army Research Office, Research Triangle Park,North Carolina. This support is gratefully acknowledged.

Appendix 1

In this appendix, the shape functions used for the beam and plate elements based on theabsolute nodal coordinate formulation are presented.

Beam element The shape function matrix S used to develop the beam element presentedin this investigation is written as follows [20]:

S = [S1I S2I S3I S4I S5I S6I S7I S8I ] (64)

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951

952

953

954

955

956

957

958

959

960

961

962

963

964

965

966

967

968

969

970

971

972

973

974

975

976

977

978

979

980

981

982

983

984

985

986

987

988

989

990

991

992

993

994

995

996

997

998

999

1000

where I is the 3 × 3 identity matrix, and the shape functions Si are defined as follows:

S1 = 1 − 3ξ 2 + 2ξ 3, S2 = l(ξ − 2ξ 2 + ξ 3

), S3 = lη(1 − ξ),

S4 = lζ(1 − ξ), S5 = 3ξ 2 − 2ξ 3, S6 = l(−ξ 2 + ξ 3

),

S7 = lηξ, S8 = lζ ξ

⎫⎪⎪⎬⎪⎪⎭ (65)

where ξ = x/l, η = y/l, ζ = z/ l, and l is the length of the element.

Plate element The shape function matrix S used to develop the plate element presented inthis investigation is written as follows [11]:

S = [S1I S2I S3I S4I S5I S6I S7I S8IS9I S10I S11I S12I S13I S14I S15I S16I ] (66)

where I is the 3 × 3 identity matrix, and the shape functions Si are defined as follows:

S1 = (2ξ + 1)(ξ − 1)2(2η + 1)(η − 1)2, S2 = lξ(ξ − 1)2(2η + 1)(η − 1)2,

S3 = wη(ξ − 1)2(2ξ + 1)(η − 1)2, S4 = hζ(ξ − 1)(η − 1),

S5 = −ξ 2(2ξ − 3)(2η + 1)(η − 1)2, S6 = lξ 2(ξ − 1)(2η + 1)(η − 1)2,

S7 = −wηξ 2(2ξ − 3)(η − 1)2, S8 = −hξζ(η − 1),

S9 = η2ξ 2(2ξ − 3)(2η − 3), S10 = −lη2ξ 2(ξ − 1)(2η − 3),

S11 = −wη2ξ 2(η − 1)(2ξ − 3), S12 = hζξη,

S13 = −η2(2ξ + 1)(ξ − 1)2(2η − 3), S14 = −lξη2(ξ − 1)2(2η − 3),

S15 = wη2(ξ − 1)2(2ξ + 1)(η − 1), S16 = −hηζ(ξ − 1)

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

(67)

where ξ = x/l, η = y/w, ζ = z/h, l is the length, w is the width and h is the thickness ofthe element.

Appendix 2

In this appendix, the transformation matrix, which is used to define the nodal coordinates inthe rotating frame, is obtained. The element nodal coordinate vector at node n is defined inthe inertial frame as follows:

en =[

rnT

(∂rn

∂x

)T (∂rn

∂y

)T (∂rn

∂z

)T]T

. (68)

This nodal coordinate vector consists of 4 three-dimensional absolute positions and gradientvectors which can be defined in the rotating frame by employing the following transforma-tion:

a = A0a (69)

where

A0 =[ cos θ − sin θ 0

sin θ cos θ 00 0 1

](70)

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1001

1002

1003

1004

1005

1006

1007

1008

1009

1010

1011

1012

1013

1014

1015

1016

1017

1018

1019

1020

1021

1022

1023

1024

1025

1026

1027

1028

1029

1030

1031

1032

1033

1034

1035

1036

1037

1038

1039

1040

1041

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1043

1044

1045

1046

1047

1048

1049

1050

and a and a are vectors defined in the inertial and rotating frames, respectively; and θ isthe angle that defines the orientation of the rotating frame with respect to the inertial frame.It follows that, when the absolute nodal coordinate formulation is used, the relationshipbetween the absolute coordinates defined in the inertial frame and the absolute coordinatesdefined in the rotating frame can be, in general, written as

e = Aq (71)

where

A =

⎡⎢⎢⎣

A0 0 . . . 00 A0 . . . 0...

.... . .

...

0 0 . . . A0

⎤⎥⎥⎦ . (72)

In this equation, A has dimension equal to the total number of nodal coordinates of thebeam. Note that for one element e with volume V e and mass density ρe, one has

AT MA =

⎡⎢⎢⎢⎣

AT0 0 . . . 0

0 AT0 . . . 0

......

. . ....

0 0 . . . AT0

⎤⎥⎥⎥⎦

×∫

V e

ρe

⎡⎢⎢⎢⎣

S21 I S1S2I . . . S1SnI

S2S1I S22 I . . . S2SnI

......

. . ....

SnS1I SnS2I . . . S2nI

⎤⎥⎥⎥⎦ dV e

⎡⎢⎢⎢⎣

A0 0 . . . 00 A0 . . . 0...

.... . .

...

0 0 . . . A0

⎤⎥⎥⎥⎦ (73)

which can be written as

AT MA =∫

V e

ρe

⎡⎢⎢⎢⎣

AT0 S2

1 A0 AT0 S1S2A0 . . . AT

0 S1SnA0

AT0 S2S1A0 AT

0 S22 A0 . . . AT

0 S2SnA0

......

. . ....

AT0 SnS1A0 AT

0 SnS2A0 . . . AT0 S2

nA0

⎤⎥⎥⎥⎦ dV e. (74)

Using the fact that A0 is an orthogonal matrix, the preceding equation shows that

AT MA = M. (75)

Similarly, one can write

AT MA =∫

V e

ρe

⎡⎢⎢⎢⎣

AT0 S2

1 A0 AT0 S1S2A0 . . . AT

0 S1SnA0

AT0 S2S1A0 AT

0 S22 A0 . . . AT

0 S2SnA0

......

. . ....

AT0 SnS1A0 AT

0 SnS2A0 . . . AT0 S2

nA0

⎤⎥⎥⎥⎦ dV e. (76)

Using the identities of (57), and the fact that A0 = θA0θ ; the preceding equation shows thatAT MA is a constant matrix if the angular velocity θ is constant, which is the case in theanalysis presented in this paper.

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1051

1052

1053

1054

1055

1056

1057

1058

1059

1060

1061

1062

1063

1064

1065

1066

1067

1068

1069

1070

1071

1072

1073

1074

1075

1076

1077

1078

1079

1080

1081

1082

1083

1084

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1086

1087

1088

1089

1090

1091

1092

1093

1094

1095

1096

1097

1098

1099

1100

Following a similar procedure, one can also write

AT MA =∫

V e

ρe

⎡⎢⎢⎢⎣

AT0 S2

1 A0 AT0 S1S2A0 . . . AT

0 S1SnA0

AT0 S2S1A0 AT

0 S22 A0 . . . AT

0 S2SnA0

......

. . ....

1AT0 SnS1A0 AT

0 SnS2A0 . . . AT0 S2

nA0

⎤⎥⎥⎥⎦ dV e. (77)

Using the identities of (57), it can be shown that the matrix AT MA is constant under theassumptions stated in this paper.

References

1. Bauchau, O.A.: Computational schemes for flexible nonlinear multibody systems. Multibody Syst. Dyn.2, 169–225 (1998)

2. Bauchau, O.A., Bottasso, C.L., Nikishkov, Y.G.: Modeling rotorcraft dynamics with finite element multi-body procedures. J. Math. Comput. Modeling 33, 1113–1137 (2001)

3. Berzeri, M., Shabana, A.A.: Study of the centrifugal stiffening effect using the finite element absolutenodal coordinate formulation. Multibody Syst. Dyn. 7, 357–387 (2002)

4. Cesnik, C.E.S., Hodges, D.H., Sutyrin, V.G.: Cross-sectional analysis of composite beams includinglarge initial twist and curvature effects. AIAA J. 34, 1913–1920 (1996)

5. Crisfield, M.A.: Non-Linear Finite Element Analysis of Solids and Structures, vol. I: Essentials. Wiley,New York (1997)

6. Garcia-Vallejo, D., Sugiyama, H., Shabana, A.A.: Finite element analysis of the geometric stiffeningeffect, part 1: a correction in the floating frame of reference formulation. IMechE J. Multibody Dyn.219, 187–202 (2004)

7. Garcia-Vallejo, D., Sugiyama, H., Shabana, A.A.: Finite element analysis of the geometric stiffeningeffect, part 2: non-linear elasticity. IMechE J. Multibody Dyn. 219, 203–211 (2004)

8. Hodges, D.H.: A review of composite rotor blade modeling. AIAA J. 28, 561–565 (1990)9. Hussein, B., Sugiyama, H., Shabana, A.A.: Coupled deformation modes in the large deformation finite

element analysis: problem definition. ASME J. Comput. Nonlinear Dyn. 2, 146–154 (2007)10. Johnson, W.: Helicopter Theory. Princeton University Press, New Jersey (1980)11. Mikkola, A.M., Shabana, A.A.: A non-incremental procedure for the analysis of large deformation of

plates and shells in mechanical systems application. Multibody Syst. Dyn. 9, 283–309 (2003)12. Pascal, M.: Some open problems in dynamic analysis of flexible multibody systems. Multibody Syst.

Dyn. 5, 315–334 (2001)13. Reissner, E.: On a variational theorem for finite elastic deformation. J. Math. Phys. 32, 129–135 (1953)14. Schilhans, M.J.: Bending frequency of a rotating cantilever beam. Trans. ASME J. Appl. Mech. 25,

28–30 (1958)15. Schwab, A.L., Meijard, J.P.: Comparison of three-dimensional flexible beam elements for dynamic

analysis: finite element method and absolute nodal coordinate formulation. In: Proceedings of ASMEInternational Design Engineering Technical Conferences and Computer Information in Engineering Con-ference (DETC2005/ MSNDC-85104), Long Beach, CA, USA (2005)

16. Shabana, A.A.: Dynamics of Multibody Systems, 3rd edn. Cambridge University Press, Cambridge(2005)

17. Shabana, A.A., Yakoub, R.Y.: Three dimensional absolute nodal coordinate formulation for beam ele-ments: Theory. ASME J. Mech. Des. 123, 606–613 (2001)

18. Wallrapp, O., Schwertassek, R.: Representation of geometric stiffening in multibody system simulation.Int. J. Numer. Methods Eng. 32, 1833–1850 (1991)

19. Wu, S.C., Haug, E.J.: Geometric non-linear substructuring for dynamics of flexible mechanical systems.J. Numerical Methods Eng. 26, 2211–2226 (1988)

20. Yakoub, R.Y., Shabana, A.A.: Three-dimensional absolute nodal coordinate formulation for beam ele-ments: implementation and applications. ASME J. Mech. Des. 123, 614–621 (2001)

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